Bousfield-Kan spectral sequence with local coefficients

Question

Let $F : \mathcal{D} \to \mathbf{Top}$ be a diagram of topological spaces. A local system of coefficients $M$ on $\mathrm{colim}_\mathcal{D} F$ pulls back to a local system $M_d$ on $F(d)$ for each $d \in \mathcal{D}$, and also a local system $M_h$ on $\mathrm{hocolim}_\mathcal{D} F$.

Is there a Bousfield-Kan type spectral sequence of the form $$E^2_{s,t} = \mathrm{colim}^s_{\mathcal{D}} H_t(F(d);M_d) \Rightarrow H_{s+t}(\mathrm{hocolim}_\mathcal{D} F;M_h)$$ and if so where can one find it in the literature? I would also be content to know if this is not possible.

Answer 1

Let LOC be the category in which an object is a space plus a local system on it, and a morphism is a map of spaces covered by a map of coefficient systems in the obvious sense. There's an obvious functor $C$ from LOC to CH, the category of chain complexes; one can speak of the hocolim of a diagram of chain complexes; and $C$ commutes with hocolim up to natural equivalence. Your setup yields a functor $\mathcal D\to LOC$, and then your problem becomes the algebraic problem of making a spectral sequence going from $E^2_{s.t}=colim_{\mathcal D}^s H_t(F)$ to $H_{s+t}(hocolim_{\mathcal D} F)$ when $F$ is a functor $\mathcal D\to CH$. For this you can, as Tyler suggests, use a simplicial model for $hocolim_{\mathcal D}F$ related to the nerve of $\mathcal D$ and thus get a two-quadrant double chain complex for which one of the two standard filtrations yields a spectral sequence of the desired kind. ($colim^i_{\mathcal D}$ means the $ith$ derived functor of the functor $colim_{\mathcal D}$ from $\mathcal D$-diagrams of abelian groups to abelian groups; it is the same as $ith$ homology of hocolim of the diagram of (abelian groups viewed as) chain complexes.)